Flow Through a Slit

Flow through a slit head has been studied also in polymers using the power-function flow principle (Ostwald-de Villes model). Some of the first works in this field were carried out in 1967-1968 by N. V. Tyabin et al.21,22). Tests were arranged with molten polyethylene in an extrusion head having a rotary conic core. The divergence between theory and experiment was comparatively significant which, apparently, is accounted for, primarily, by the basence of a necessary similarity between the conducted experiment and the theoretical description of the flow in a flat slit. 

Figure 5.12 Schematic diagram of pressure flow through a slit.

Derive the equations for pressure driven flow through a slit using a shear thinning power law viscosity model. 

It is interesting to point out that for a die with a pressure flow through a slit, or sets of slits, the optimum channel depth is directly proportional to the die gap. Decreasing the die gap by a certain percentage will result in an optimum channel depth that is reduced by the same percentage. To determine the optimum helix angle we can re-write eqn. (6.23) for this specific application,... 

Figure 3. Navier-Stokes flow through a slit geometrical dimensions and zoom of finite element mesh.

A fluid obeying a power law given by cr = [see Eq. (13.37)] with viscosity 250 N s/m, flows through a slit channel of narrow rectangular section, driven by a pressure drop per unit length of AP/AL = 2.5 x 10 Pa/ m. The channel has h = 2 mm thickness, and its width is w = 20 cm. Find an equation giving the velocity profile into the channel and the velocity gradient at the wall as a function of the flow rate 0. 

Another potymer melt is extruded through the tube die described in Problem 7.12. This melt may be assumed to obey the power law, with viscosities 4500 and 1000 Pas at shear rates 10 and 100 s respectively. A pressure drop per unit length of 10 Pam is applied to the die. Find the volume flow rate at which the tube is extruded. (A useful equation for power-law flow through a slit-like chaimel is given in Problem 7.11.)... 

In additional to cylindrical capillary tubes, another type of capillary with a simple cross-sectional shape is the slit capillary, i.e., a channel formed between two parallel plates. For an electrolyte solution flowing through a slit capillary, it can be shown that, in analogy to Eq. 13,... 

A Newtonian fluid, of viscosity flows through a slit-like channel of narrow rectangular section, driven by a pressure drop per unit length AP/AL. The channel is of depth h and width (transverse to the flow) w, where w> h. 

The schematic of synthetic jet is shown in Fig. 2. It consists of a cavity open to the flow through a slit or orifice. The bottom wall of the cavity is driven by a moving surface, which can be either a piston or a vibrating membrane. The vibration of the membrane can be achieved using electrostatic... 

Two limiting incompressible flow cases for the annulus are represented by flow through a slit and flow through a circular tube. The former is for the case of a large core (i.e., inner tube), whereas the latter represents a small core. For a 0.0254 m tube find the range of cores for each case that will be within 15 percent of the pressure drop values for the annulus. 

Pressure-driven axial flow through a narrow annulus is essentially the same as flow through a slit, but without the side walls. The lack of side walls may be helpful for studying slip phenomena and for reducing residence time distribution. Furthermore, the pressure difference between the outer and inner walls of the annulus gives the second normal stress difference. 

We now present the theory (Davis et al. 1973 Han 1974) that allows one to determine shear stress and first normal stress difference in steady-state shear flow using wall normal stress measurements along the axis of a slit die. Consider a fluid flowing through a slit die having the height h and the width w, and assume that flow has become fully developed. Then, for steady-state fully developed flow, the equations of motion... 

When a very viscous molten polymer is forced to flow through a slit or capillary die, viscous shear heating can become significant above a certain critical value of y or ct. Under such situations, nonlinear profiles of wall normal stress in a slit or capillary die may be observed, as described in the preceding section. Therefore, continuous-flow capillary/slit rheometry is limited to y or a, below which viscous shear heating can be neglected. 

Methods for determining the first normal stress difference at higher shear rates than are accessible using a cone-plate rheometer include the Lodge Stressmeter rheometer [117-119] and the sliding plate rheometer [84]. The former makes use of the pressure measured at the bottom of a small side hole in flow through a slit. It has been demonstrated that these two instruments give comparable results for a polystyrene melt up to shear rates of 200 s" [85]. 

B.4 Adapting the Parallel Plate Solution to Annular Flow. For small annular gaps (e.g., k = 0.9) the expression for Q for flow through a slit of a power-law fluid (Problem 2B.3) ean be used to obtain an expression for Q for annular flow. Adapt the parallel plate flow solution for the power-law model to that for flow through an annulus with a small gap (you should obtain the expression in Eq. 2.103). 

B.9 Multiple Layer Flow Through a Slit Die. Obtain expressions for the flow of three fluids through a slit die using the notation in Figure 7.51. In particular, find the velocity field and the volumetric flow rate for each layer. Confirm your solution by comparing with the expressions in Eqs. 7.60 and 7.61. 

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